Warning: Wikipedia contains spoilers
Sideshow Bob is highly educated and intelligent. Angered by being overshadowed by Krusty's "dilapidated vaudeville antics", not to mention Krusty's continual abuse of Bob, he snapped and framed Krusty for armed robbery. For a while he took over Krusty's place and changed the show, making it more upmarket (for example, Krusty's catchphrase "Children, who do you love?" was replaced by the more grammatically correct "Whom do you love?"). His reign was, however, shortlived, for Bart exposed his plan and landed him in jail.
Since then, he has taken the position of Evil Genius in the series - a typical Sideshow Bob episode involves his being released from incarceration and acting out some cunning plan, which is foiled by Bart and (usually) Lisa. Frequently this plan involves killing someone: Krusty, Bart or everyone in Springfield.
Bob was briefly married to Selma (whom he also tried to kill), and was the Robert Mitchum/Robert de Niro character in the Cape Fear sendup episode. He is a Republican, and was briefly mayor of Springfield. He can sing entire Gilbert and Sullivan scores off the cuff.
Bob also has several tattoos, such as "Die Bart Die" (German, he claims, for "the Bart, the") on his chest and one resembling Bart's decapitated head on a skateboard on his back. The tattoos on Bob's fingers parody those of Robert Mitchum's character in the movie The Night of the Hunter (whose fingers are tattooed L-O-V-E and H-A-T-E), but, as Simpsons' characters have only three fingers, Bob's finger tattoos spell "L-U-V" and "H-Ā-T".
His position as Krusty's sidekick is now taken by Sideshow Mel.
Grammer's Frasier costar David Hyde Pierce made an appearance in one episode as Sideshow Bob's brother Cecil.